Just a couple of thoughts:

If you take the tire and rim off the buggy, and let all the air out of the tire, does the sidewall completely collapse? If not, then the sidewall does carry some of the load.

Second, if what you are saying about the footprint expanding as the load increases is true, then there should be absolutely no difference in damage taking a fully loaded cart across the field versus an empty one. I've never talked to anyone who believed that. If psi is the only factor in field damage, then my bicycle would cause far more damage to your field than a fully loaded grain cart.